Solved examples: finding limits from graphs with detailed solutions

📚 Quick Study Guide

• 🔍 Limit Definition: The limit of a function $f(x)$ as $x$ approaches $c$ is $L$, written as $\lim_{x \to c} f(x) = L$, if $f(x)$ gets arbitrarily close to $L$ as $x$ gets arbitrarily close to $c$ (but not necessarily equal to $c$).
• 📈 Graphical Interpretation: Visually, the limit is the y-value that the function approaches as you trace the graph from both the left and right sides toward $x = c$.
• ⛔ One-Sided Limits:
  • ➡️ Right-Hand Limit: $\lim_{x \to c^+} f(x) = L$ means the limit as $x$ approaches $c$ from the right is $L$.
  • ⬅️ Left-Hand Limit: $\lim_{x \to c^-} f(x) = L$ means the limit as $x$ approaches $c$ from the left is $L$.
• ✅ Existence of a Limit: For the limit $\lim_{x \to c} f(x)$ to exist, both one-sided limits must exist and be equal: $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$.
• 💔 Discontinuities: Pay attention to holes, jumps, and vertical asymptotes in the graph. These often indicate where limits might not exist or might be infinite.
• 📍 Holes: If there's a hole at $x=c$, the limit might still exist if the function approaches the same y-value from both sides.
• 🪜 Jump Discontinuities: If the function jumps at $x=c$, the left-hand and right-hand limits will be different, and the overall limit does not exist.
• 📏 Vertical Asymptotes: If the function approaches infinity (or negative infinity) as $x$ approaches $c$, the limit does not exist. We often denote these as infinite limits.

Practice Quiz

1. Given a graph of $f(x)$, if $\lim_{x \to 2^-} f(x) = 3$ and $\lim_{x \to 2^+} f(x) = 3$, what is $\lim_{x \to 2} f(x)$?

   1. 3
   2. Does Not Exist
   3. 6
   4. 0

2. If a graph has a jump discontinuity at $x = -1$, what can you conclude about $\lim_{x \to -1} f(x)$?

   1. The limit exists and is equal to the average of the left and right limits.
   2. The limit exists and is equal to the value of $f(-1)$.
   3. The limit does not exist.
   4. The limit is infinite.

3. Consider a graph where $f(x)$ approaches infinity as $x$ approaches 5. What is $\lim_{x \to 5} f(x)$?

   1. 0
   2. $\infty$
   3. 5
   4. Does Not Exist

4. From the graph, it's observed that as $x$ approaches 4 from the left, $f(x)$ approaches 2. As $x$ approaches 4 from the right, $f(x)$ also approaches 2. What is $\lim_{x \to 4} f(x)$?

   1. 4
   2. 2
   3. Does Not Exist
   4. 0

5. A graph has a hole at the point $(3, 1)$, but the function approaches $y=1$ from both sides as $x$ approaches 3. What is $\lim_{x \to 3} f(x)$?

   1. The limit does not exist because there is a hole.
   2. The limit is 1.
   3. The limit is 3.
   4. The limit is undefined.

6. If $\lim_{x \to 0^-} f(x) = -1$ and $\lim_{x \to 0^+} f(x) = 1$, what can be concluded about $\lim_{x \to 0} f(x)$?

   1. The limit exists and is 0.
   2. The limit exists and is 1.
   3. The limit exists and is -1.
   4. The limit does not exist.

7. Given a graph, as $x$ approaches -2 from both sides, $f(x)$ approaches 0. However, $f(-2)$ is defined as 5. What is $\lim_{x \to -2} f(x)$?

   1. 5
   2. 0
   3. -2
   4. Does Not Exist